Triangle XYZ

Question

is drawn on a coordinate plane with vertices X(0,0)
, Y(3,6)
, and Z(4,3)
and with medians indicating the midpoint of each line XY
, YZ
, and ZX
. Prove that the medians meet at a single point by finding the centroid.(1 point)
centroid=
(
,
)

Answer 1

To find the centroid of triangle XYZ with vertices \( X(0,0) \), \( Y(3,6) \), and \( Z(4,3) \), we use the formula for the centroid \( G \) of a triangle given its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \):

\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]

Substituting the coordinates of the vertices:

• \( X(0,0) \): \( (x_1, y_1) = (0, 0) \)
• \( Y(3,6) \): \( (x_2, y_2) = (3, 6) \)
• \( Z(4,3) \): \( (x_3, y_3) = (4, 3) \)

Now we calculate the \( x \)-coordinate of the centroid:

\[ x_G = \frac{0 + 3 + 4}{3} = \frac{7}{3} \]

Next, we calculate the \( y \)-coordinate of the centroid:

\[ y_G = \frac{0 + 6 + 3}{3} = \frac{9}{3} = 3 \]

Thus, the coordinates of the centroid \( G \) are:

\[ G = \left( \frac{7}{3}, 3 \right) \]

Therefore, the centroid of triangle XYZ is at:

\[ \text{centroid} = \left( \frac{7}{3}, 3 \right) \]